For \(x\in \partial M\) , let \(E_1(x), E_2(x) \in T_{x}\partial M\)  be the unit eigenvectors of the shape operator at \(x\)  corresponding respectively to the principal curvatures \(\lambda _1(x),\ \lambda _2(x)\) . We will drop the dependence in \(x\)  from our notation when there is no ambiguity. We choose \(E_1\)  and \(E_2\)  such that \(E_1^\flat \wedge E_2^\flat \wedge \nu ^\flat \)  is a positive multiple of the volume form \( d_g\)  (see p.26 of [1]} for the “musical isomorphism” notation of \(^\flat \)  and \(^\sharp \) ). Here we use \(\nu \)  to denote the outward pointing normal vector field. By \(H(x)\) , we denote the mean curvature of \(\partial M\)  at \(x\) . We also set
\({\rm {II}}_x(V) := {\rm {II}}_x(V,V),\ \ V\in T_x\partial M\) 
